MILAN, Italy (AP) — The Italian appeals court that overturned Amanda Knox's murder conviction in the slaying of her British roommate gave the reasons for its ruling on Thursday: the evidence that had been used by a lower court against the American and her Italian boyfriend just didn't hold up.

Those shortcomings included no murder weapon, faulty DNA, an inaccurate time for the killing, and insufficient proof that Knox and Raffaele Sollecito were even at the location where the crime occurred. So said the Perugia appellate court in its long-awaited reasoning behind its October ruling that reversed the lower court's convictions.

British college student Meredith Kercher was found slain in a pool of blood on her bedroom floor in Perugia, Italy, on Nov. 2, 2007.

Knox and Sollecito, who had just begun dating at the time of the murder, were arrested several days later, then convicted in what prosecutors' portrayed as a drug-fueled sexual assault. They were sentenced to 26 years and 25 years, respectively, in proceedings that made headlines around the world.

On Thursday, the appellate cited among the other failed elements of the prosecutors' case DNA evidence, which was undermined during a re-examination in the appeals trial, and the failure to conclusively identify the murder weapon. The appellate court even contradicted the lower court's time of death, saying it happened at around 10:15 p.m., not after 11 p.m. The court said the "building blocks" used to construct the case had failed.

The appeals court also said there was no proof of the prosecutors' claim that Knox and Sollecito had helped a third man, who was convicted separately, of sexually assault Kercher, nor was there evidence that the pair had simulated a burglary by throwing a rock through a window to remove suspicion from themselves, as prosecutors alleged.

The appeals court said the lower court had arrived at a verdict "that was not corroborated by any objective element of evidence and in itself was not, in fact probable: the sudden choice of two young people, good and open to other people, to do evil for evil's sake, just like that, without another reason."

"It is not, therefore, sufficient that the probability of the prosecutors' hypothesis is greater than the hypothesis of the defense, not even when they are notably greater in number, but it is necessary that every explanation that differs from the prosecutors' hypothesis is, according to the criteria of reasonability, not at all plausible," the court said.

The only elements of the prosecutors case that were proven, the appeals court said, were the charge of slander against Knox, who was convicted of falsely accusing a bar owner of killing Kercher, and the fact that the Knox and Sollecito alibis did not match.

That the alibis were out of synch "is very different" from the prosecutors' claim of false alibis, the court said.

The proven elements combined, the court said, are not enough to support convictions against Knox and Sollecito.

"The only elements that are sustained don't allow the belief, even when put together, that the guilt of Amanda Knox and Raffaele Sollecito for the crime of murder ... has been proven," the court said.

After her conviction was thrown out, Knox, 24, returned immediately home to Seattle. She was credited with time served for the conviction of slander for accusing bar owner Diya "Patrick" Lumumba of carrying out the killing.

Prosecutors contended a kitchen knife found at Sollecito's house was the weapon because it matched wounds on Kercher's body and carried traces of Kercher's DNA on the blade and Knox's on the handle. However, the court-ordered review discredited the DNA evidence, saying there were glaring errors in evidence-collecting and that below-standard testing and possible contamination raised doubts over the DNA traces on the blade and on Kercher's bra clasp.

In addition, the defense cast doubt on the knife, questioning why Knox and Sollecito would return it to Sollecito's home if it had been used in the murder. They maintain the real weapon has yet to be found.

A third defendant in the case, Rudy Hermann Guede of the Ivory Coast, was convicted in a separate trial of sexually assaulting and stabbing Kercher. His 16-year prison sentence — reduced on appeal from an initial 30 years — was upheld by Italy's highest court in 2010.

The appeals court also expressed incredulity that the two would have cooperated in such a crime with Guede, with whom there is no proof of any relationship. "For example, there is no evidence of phone calls or text messages between the three," the court said.